In this article, we present an algorithm to efficiently evaluate the exchange matrix in periodic systems when a Gaussian basis set with pseudopotentials is used. The usual algorithm for evaluating exchange matrix scales cubically with the system size because one has to perform O(N²) fast Fourier transform (FFT). Here, we introduce an algorithm that retains the cubic scaling but reduces the prefactor significantly by eliminating the need to do FFTs during each exchange build. This is accomplished by representing the products of Gaussian basis function using a linear combination of an auxiliary basis the number of which scales linearly with the size of the system. We store the potential due to these auxiliary functions in memory, which allows us to obtain the exchange matrix without the need to do FFT, albeit at the cost of additional memory requirement. Although the basic idea of using auxiliary functions is not new, our algorithm is cheaper due to a combination of three ingredients: (a) we use a robust pseudospectral method that allows us to use a relatively small number of auxiliary basis to obtain high accuracy; (b) we use occ-RI exchange, which eliminates the need to construct the full exchange matrix; and (c) we use the (interpolative separable density fitting) ISDF algorithm to construct these auxiliary basis sets that are used in the robust pseudospectral method. The resulting algorithm is accurate, and we note that the error in the final energy decreases exponentially rapidly with the number of auxiliary functions.

中文翻译：

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使用鲁棒伪谱方法与高斯基组进行快速交换

在本文中，我们提出了一种算法，可以在使用具有赝势的高斯基组时有效地评估周期性系统中的交换矩阵。评估交换矩阵的常用算法与系统大小成立方关系，因为必须执行O ( N ²) 快速傅立叶变换 (FFT)。在这里，我们介绍了一种算法，它保留了立方尺度，但通过消除在每次交换构建期间进行 FFT 的需要，显着减少了预因子。这是通过使用辅助基的线性组合来表示高斯基函数的乘积来实现的，辅助基的数量与系统的大小成线性比例关系。由于这些辅助功能，我们将势能存储在内存中，这使我们无需进行 FFT 即可获得交换矩阵，尽管以额外的内存要求为代价。虽然使用辅助函数的基本思想并不新鲜，但由于结合了三个要素，我们的算法更便宜：(a) 我们使用稳健的伪谱方法，允许我们使用相对较少的辅助基础来获得高精度；(b) 我们使用 occ-RI 交换，无需构建完整的交换矩阵；(c) 我们使用（插值可分离密度拟合）ISDF 算法来构造这些用于鲁棒伪谱方法的辅助基组。由此产生的算法是准确的，我们注意到最终能量的误差随着辅助函数的数量呈指数级快速下降。